MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnrmtop Structured version   Visualization version   GIF version

Theorem pnrmtop 22715
Description: A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
pnrmtop (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Proof of Theorem pnrmtop
StepHypRef Expression
1 pnrmnrm 22714 . 2 (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
2 nrmtop 22710 . 2 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
31, 2syl 17 1 (𝐽 ∈ PNrm → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Topctop 22265  Nrmcnrm 22684  PNrmcpnrm 22686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-cnv 5645  df-dm 5647  df-rn 5648  df-iota 6452  df-fv 6508  df-ov 7364  df-nrm 22691  df-pnrm 22693
This theorem is referenced by:  pnrmopn  22717
  Copyright terms: Public domain W3C validator